Vector Analysis

Spherical to rectangular coordinates

x = r sinq cosf
y = r sinq sinq
z = r cosq

Rectangular to spherical coordinates

r = sqrt(x²+ y² + z²)
q = cos ^-1(z/r)
f = tan ^-1(y/x) = sin ^-1(y/rsinq) = cos ^-1(x/rsinq)

If,
A = a_xi + a_yj + a_zk
B = b_xi + b_yj + b_zk ,
where i, j, and k are unit vectors in the x, y, and z directions.
|A| = sqrt(a_x² + a_y² + a_z²)
|B| = sqrt(b_x² + b_y² + b_z²)
then:

Dot (scaler) procuct

A^.B = |A||B|cos a, where a is angle between the vectors.
A^.B = a_xb_x + a_yb_y + a_zb_z

Cross Product

AXB = |A||B|sin a v, where v is a unit vector perpendicular to A and B
AXB = (a_yb_z - a_zb_y) i + (a_zb_x - a_xb_z) j + (a_xb_y - a_yb_x) k